There are a lot of FReepers whose ideas run precisely along those lines. The practicality of rounding up illegals, and the political and legal realities of dealing with the people so-collected, are lost on the border-bots.
You are quite correct that the problem originates among those who are paying illegals to come here. I would extend the blame to various state, local, and federal entities that provide long-term welfare benefits to those folks.
It’s a simple fix. Turn off the faucet of freebies, and amazingly they leave on their own.